Too Risk Averse to Purchase Insurance? A
Theoretical Glance at the Annuity Puzzle
Antoine Bommier, Francois Le Grand
ETH Risk Center – Working Paper Series
The ETH Risk Center, established at ETH Zurich (Switzerland) in 2011, aims to develop cross-
disciplinary approaches to integrative risk management. The center combines competences from the
natural, engineering, social, economic and political sciences. By integrating modeling and simulation
efforts with empirical and experimental methods, the Center helps societies to better manage risk.
More information can be found at: http://www.riskcenter.ethz.ch/.
Too Risk Averse to Purchase Insurance? A Theoretical Glance at
the Annuity Puzzle
Antoine Bommier, Francois Le Grand
This paper suggests a new explanation for the low level of annuitization, which is valid even if one as-
sumes perfect markets. We show that, as soon there exists a positive bequest motive, sufficiently risk
averse individuals should not purchase annuities. A model calibration accounting for temporal risk aver-
sion generates a willingness-to-pay for annuities, which is significantly smaller than the one generated by
a standard Yaari (1965) model. Moreover, the calibration predicts that riskless savings finances one third
of consumption, in line with empirical findings. Keywords: annuity puzzle, insurance demand, bequest,
intergenerational transfers, temporal risk aversion, multiplicative preferences.
Keywords: annuity puzzle, insurance demand, bequest, intergenerational transfers, temporal risk aver-
sion, multiplicative preferences
Classifications: JEL Codes: D11, D81, D91.
URL: http://web.sg.ethz.ch/ethz risk center wps/ETH-RC-12-002
Notes and Comments:
ETH Risk Center – Working Paper Series
Too Risk Averse to Purchase Insurance?
A Theoretical Glance at the Annuity Puzzle∗
Antoine Bommier François Le Grand†
January 23, 2012
This paper suggests a new explanation for the low level of annuitization, which is valid
even if one assumes perfect markets. We show that, as soon there exists a positive bequest
motive, sufficiently risk averse individuals should not purchase annuities. A model calibration
accounting for temporal risk aversion generates a willingness-to-pay for annuities, which is
significantly smaller than the one generated by a standard Yaari (1965) model. Moreover,
the calibration predicts that riskless savings finances one third of consumption, in line with
Keywords: annuity puzzle, insurance demand, bequest, intergenerational transfers, tem-
poral risk aversion, multiplicative preferences.
JEL codes: D11, D81, D91.
Among the greatest risks in life is that associated with life expectancy. A recently retired American
man of age 65 has a life expectancy of about 17.5 years. Though, there is a more than 22% chance
that he will die within the first 10 years and a more than 20% chance that he will live more than
25 years. Savings required to sustain 10 or 25 years of retirement vary considerably, and one would
expect a strong demand for annuities, which are financial securities designed to deal with such a
∗We are grateful to Edmund Cannon, Alexis Direr, Lee Lockwood, Thomas Post, James Poterba, Ray Rees,
Harris Schlesinger and seminar participants at ETH Zurich, University of Paris I, University of Zurich, 2011 Summer
Meetings of the Econometric Society for their comments.
†Antoine Bommier is Professor at ETH Zurich; firstname.lastname@example.org François Le Grand is Assistant Professor at
EMLyon Business School and Associate Researcher at ETH Zurich; email@example.com. Both authors gratefully
acknowledge financial support from Swiss-Re.
lifetime uncertainty. A number of papers have stressed the utility gains that would be generated
by the annuitization of wealth at retirement. It is generally estimated that individuals would be
willing to give up to 25% of their wealth at retirement to gain access to a perfect annuity market
(see Mitchell, Poterba, Warshawsky and Brown (1999) among others). According to standard
theoretical predictions, even when individuals have a bequest motive, they should fully annuitize
the expected value of their future consumption. However, puzzlingly enough, empirical evidence
consistently shows that individuals purchase very few private annuities, in sharp contradiction
with the theoretical predictions. For example, Johnson, Burman, and Kobes (2004) report than
in the US, private annuities finance less than 1% of household income for people older than 65.1
Similarly, they also observe that private annuities are only purchased by 5% of people older than
65. James and Song (2001) find similar results for other countries, such as Canada, the United
Kingdom, Switzerland, Australia, Israel, Chile and Singapore.
A number of explanations to this puzzle have been suggested, relying on market imperfections
or rationality biases.2 For example, due to imperfect health insurance, individuals would need
to store a substantial amount of liquidities; unfair annuity pricing would make them unattractive
assets; or framing effects would play an important role in agents’ decisions to annuitize.
In this paper we emphasize that, even if the annuity market were perfect, a low (or even zero)
level of annuitization can be fully rational. Our explanation relies on the role of risk aversion.
We show that a high level of risk aversion together with a positive bequest motive is sufficient to
predict a negative demand for annuities.
The reason why this explanation remained unexplored is that the literature has mainly focused
on time additively separable preferences, or on Epstein and Zin specification, while both models
are unadapted to study the role of risk aversion (See Bommier, Chassagnon, LeGrand (2010),
henceforth BCL). In the current paper, the role of risk aversion is investigated in the expected
utility framework, through the concavification of the lifetime utility function as introduced by
Kihlstrom and Mirman (1974). We prove that the demand for annuities decreases with risk aversion
and eventually vanishes when risk aversion is large enough.
The fact that annuity demand decreases – and does not increase – with risk aversion might
seem counterintuitive. Insurance demand is generally found to increase with risk aversion. Though,
this correlation does not hold when irreplaceable commodities, such as life, are at risk. As was
explained by Cook and Graham (1977), rational insurance decisions aim at equalizing marginal
utilities of wealth across states of nature. With irreplaceable commodities, this may generate risk
1Roughly one half of income stems from public pensions, 17% from firm sponsored pension payments and one
third is financed from savings.
2See Brown (2007), as well as the following section for a literature review.
taking behavior. Whenever this is the case, one should expect risk aversion to decrease risk taking
and reduce the demand for insurance.3
Annuities provide an example where purchasing insurance is risk increasing. Lifetime is un-
certain, but living long is generally considered to be a good outcome, while dying early is seen
to be a bad outcome. For a given amount of savings, purchasing annuities, rather than bonds
for example, involves reducing bequest in the case of an early death (i.e., a bad outcome), while
increasing consumption in case of survival (i.e., a good outcome). Thus, for a given level of savings,
annuities transfer resources from bad to good states of the world and are, as such, risk increasing.
If first period consumption were exogenous and inter vivos transfers ruled out, simple dominance
arguments as in BCL would directly imply that the demand for annuity decreases with risk aver-
sion. In the current paper, the result is obtained with endogenous consumption smoothing and the
introduction of inter vivos transfers. Moreover, we prove that when risk aversion is large enough,
annuity demand eventually vanishes.
In order to evaluate to what extent risk aversion contributes to solving the annuity puzzle, we
calibrate a life-cycle model in which agents can invest in bonds and annuities. Calibrating risk
aversion and bequest motives to plausible levels shows that risk aversion alone does not generate
a negative demand for annuities. However, we obtain considerably smaller willingnesses-to-pay
for annuities than those obtained with the standard Yaari model, indicating that risk aversion
may indeed be an important factor to explain the low levels of annuitization. Our calibration
implies that one third of the agents’ consumption is financed by riskless savings, which is in line
with empirical findings of Johnson, Burman, and Kobes (2004). This contrasts with the standard
Yaari’s model in which riskless savings do not contribute at all to consumption financing, even if
agents have bequest motives.
The remainder of the paper is structured as follows. In Section 2, we discuss the related
literature. We then present a two-period model and derive our theoretical predictions in Section 3.